We consider a deformation of the two-dimensional free scalar field theory by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it is no longer invariant under the full Witt algebra. Furthermore, there is an adelic version of this family of conformal field theories, parameterized by the choice of a number field, together with a Hecke character. Tate's thesis gives the Green's functions of these theories, and ensures that these Green's functions satisfy an adelic product formula. In particular, the local L-factors contribute to the prefactors of these Green's functions. Quadratic reciprocity turns out to be a consequence of an adelic version of a holomorphic factorization property of this family of theories on a quadratic extension of Q. We explain that at the Archimedean place, the desired holomorphic factorization follows from the global conformal symmetry.

We consider a p2-brane model as a theory of maps from the vertices of the Bruhat-Tits tree times Z into R^d. We show that in order for the worldsheet time evolution to be unitary, a certain spectral parameter of the model must localize at special loci in the complex plane, which include the (0,1) interval and the critical line of real part 1/2. The excitations of the model are supported at these loci, with commutation relations closely resembling those of the usual bosonic string. We show that the usual Hamiltonian, momentum, and angular momentum are conserved quantities, and the Poincaré algebra is obeyed. Assuming an Euler product relation for the spectrum, the Riemann zeta zeros on the critical line have spectral interpretation as the massless photon and graviton in the Archimedean theory.

Given a number field K with a Hecke character \chi, for each place \nu we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of \chi. These theories appear in the study of p-adic string theory and p-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of \chi. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places \nu, the field theory two-point functions corresponding to the Green's functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate's thesis in adelic physics.

For an arbitrary prime number p, we propose an action for bosonic p-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum p-adic strings, similar to the ordinary bosonic strings case. It turns out that spherical vectors of unramified principal series representations of PGL(2,Qp) are the plane wave modes of the bosonic fields on p-adic strings, and that the regularized normalization of these modes on the p-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.

I propose that physical theories defined over finite places (including p-adic fields) can be used to construct conventional theories over the reals, or conversely, that certain theories over the reals "decompose" over the finite places, and that this decomposition applies to quantum mechanics, field theory, gravity, and string theory, in both Euclidean and Lorentzian signatures. I present two examples of the decomposition: quantum mechanics of a free particle, and Euclidean two-dimensional Einstein gravity. For the free particle, the finite place theory is the usual free particle p-adic quantum mechanics, with the Hamiltonian obtained from the real one by replacing the usual derivatives with Vladimirov derivatives, and numerical coefficients with p-adic norms. For Euclidean two-dimensional gravity, the finite place objects mimicking the role of the spacetime are SL(Qp) Bruhat-Tits trees. I furthermore propose quadratic extension Bruhat-Tits trees as the finite place objects into which Lorentzian AdS2 decomposes, and Bruhat-Tits buildings as a natural generalization to higher dimensions, with the same symmetry group on the finite and real sides for the manifolds and buildings corresponding to the vacuum state. I comment on the implications of this decomposition for the cosmological constant problem, black hole information loss problem, and construction of black hole microstates.

Melonic field theories are defined over the p-adic numbers with the help of a sign character. Our construction works over the reals as well as the p-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory.

We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph should either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with p-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.

In this paper we argue that classical, asymptotically AdS spacetimes that arise as states in consistent ultraviolet completions of Einstein gravity coupled to matter must satisfy an infinite family of positive energy conditions. To each ball-shaped spatial region B of the boundary spacetime, we can associate a bulk spatial region Σ_B between B and the bulk extremal surface B̃ with the same boundary as B. We show that there exists a natural notion of a gravitational energy for every such region that is non-negative, and non-increasing as one makes the region smaller. The results follow from identifying this gravitational energy with a quantum relative entropy in the associated dual CFT state. The positivity and monotonicity properties of the gravitational energy are implied by the positivity and monotonicity of relative entropy, which holds universally in all quantum systems.

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit T→∞, and a universal spectrum in the Cardy regime, Δ→∞. We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all Δ≥c/6. The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal `enigma' saddlepoints in the thermodynamics of 3d gravity.

We generalize our holographic derivation of spontaneous angular momentum generation in 2 + 1 dimensions in several directions. We consider cases when a parity violating perturbation responsible for the angular momentum generation can be non-marginal (while in our previous paper we restricted to a marginal perturbation), including all possible two-derivative interactions, with parity violations triggered both by gauge and gravitational Chern-Simons terms in the bulk. We make only a minimal assumption about the bulk geometry that it is asymptotically AdS, respects the Poincaré symmetry in 2 + 1 dimensions, and has a horizon. In this generic setup, we find a remarkably concise and universal formula for the expectation value of the angular momentum density, to all orders in the parity violating perturbation.